Intereting Posts

For which categories we can solve $\text{Aut}(X) \cong G$ for every group $G$?
$PGL(n, F)=PSL(n, F)$
The link between vectors spaces ($L^2(-\pi, \pi$) and fourier series
Ideals in a non-dedekind domain that cannot be factored into product of primes
Probability of ball ownership
Is every linear ordered set normal in its order topology?
Denominator in rational gcd of integer polynomials
Integrate $ \int_0^\infty ( x A+I)^{-1} A -I \frac{1}{c+x} dx $ where $A$ is psd and $c>0$
What is wrong with this deduction of $\text{ZF} \vdash \text{Cons ZF}$
Evaluate $\sum_{n=1}^\infty \frac{n}{2^n}$.
A group of order $595$ has a normal Sylow 17-subgroup.
Counting edges in a specially defined graph
Eigenvalues of doubly stochastic matrices
How Many Symmetric Relations on a Finite Set?
The meaning of implication in logic

A horse has a rubber band attached to it which can expand infinitely and is tied to a pole on the other end. At first the length of the rubber band is $l$. on the pole-side of the rubber band there is a snail. If both start walking at the same time: The horse at speed $u$ and the snail at speed $v$ with $u>v$ when will the snail catch the horse?

- Rotation of complex numbers in a complex plane. Check my work?
- If $f$ is nonnegative and continuous on $$, then $\left(\int_a^b f(x)^n \ dx\right)^{1/n}\to\max\limits_{} f$
- Finding local maximum and minimum
- Why does this infinite series equal one?
- What is the integral of 0?
- Integral eigenvectors and eigenvalues
- Show that $\int_1^{\infty } \frac{(\ln x)^2}{x^2+x+1} \, dx = \frac{8 \pi ^3}{81 \sqrt{3}}$
- Infinitesimals - what's the intuition?
- Are Continuous Functions Always Differentiable?
- Proving $\,f$ is constant.

The length of the rubber band at time $t$ is $l(t)=l+vt$. The fraction of the total band covered by the snake in one second at time $t$ is $\frac u{l+vt}$. We are going to integrate this from $t=0$ to $t=\infty$. If the integral is less than $1$, the snail will never reach the horse. If it is $1$ or larger, it will reach the horse. Since we have

$$

\int\frac u{l+vt}\,dt=\frac uv \ln(l+vt)

$$

and $\lim_{t\to \infty}\log(1+vt)=\infty$, the snail will catch the horse. This happens at time $t=T$ for which

$$

\int_{t=0}^{t=T}\frac u{l+vt}\,dt=1\\

\frac uv(\ln(l+vT)-\ln (l))=1\\

\ln\left(\frac{l+vT}l\right)=\frac vu\\

T=\frac{l(e^{\frac vu}-1)}v

$$

- Symmetrical and skew-symmetrical part of rotation matrix
- Weak Formulations and Lax Milgram:
- Some co-finite subsets of rational numbers
- How to prove $\cos 36^{\circ} = (1+ \sqrt 5)/4$?
- covariant and contravariant components and change of basis
- Measure over intersection of set with another measure
- Non-integrability of distribution arising from 1-form and condition on 1-form
- column space of positive semidefinite matrix
- Given dividend and divisor, can we know the length of nonrepeating part and repeating part?
- Show $\lim_{N\to \infty}\sum_{k=1}^{N}\frac{1}{k+N}=\ln(2)$
- how to integrate $\frac{1}{\sqrt{e^{2x}+e^x+1}}$
- How would I prove $|x + y| \le |x| + |y|$?
- What automorphisms exist on the abelian group of positive rationals under multiplication?
- For a Noetherian ring $R$, we have $\text{Ass}_R(S^{-1}M)=\text{Ass}_R(M)\cap \{P:P\cap S=\emptyset\}$
- Evaluate $\int_0^{\pi/2}\log\cos(x)\,\mathrm{d}x$